so that E
〈DF,−DL
−1
F 〉
ℓ
2
Z
− VarF
2
= E
2J
2
g ⋆
1 1
g 1
∆
2
+ 3J
1
f ⋆
1 1
g
2
= 8
kg ⋆
1 1
g 1
∆
2
k
2 ℓ
2
Z
⊗2
+ 9k f ⋆
1 1
g k
2 ℓ
2
Z
. Hence,
B
1
6 q
8 kg ⋆
1 1
g 1
∆
2
k
2 ℓ
2
Z
⊗2
+ 9k f ⋆
1 1
g k
2 ℓ
2
Z
6 2
p 2
kg ⋆
1 1
g 1
∆
2
k
ℓ
2
Z
⊗2
+ 3k f ⋆
1 1
g k
ℓ
2
Z
. Now, let us consider B
2
. We have D
k
F 6 |f k| + 2
X
i ∈Z
|gi, k|. Similarly,
D
k
L
−1
F =
f k + J
1
g ·, k
= f k +
X
i ∈Z
gi, kX
i
6 | f k| +
X
i ∈Z
|gi, k| 6 | f k| + 2 X
i ∈Z
|gi, k|. Still using a + b
4
6 8a
4
+ b
4
, we deduce X
k ∈Z
D
k
L
−1
F ×
D
k
F
3
6 X
k ∈Z
| f k| + 2 X
i ∈Z
|gi, k|
4
6 8
X
k ∈Z
f
4
k + 16 X
i ∈Z
|gi, k|
4
. Hence
B
2
6 160
3 X
k ∈Z
f
4
k + 16 X
i ∈Z
|gi, k|
4
and the desired conclusion follows by applying Theorem 3.1.
5.3 Bounds for infinite 2-runs
When d = 1, Proposition 5.1 allows to deduce the following bound for the normal approximation of the random variable F
n
defined in 5.51.
Proposition 5.3. Let {F
n
: n 1 } be the sequence defined by F
n
=
G
n
−EG
n
p
VarG
n
with G
n
= X
i ∈Z
α
n i
ξ
i
ξ
i+1
.
1728
Here, ξ = {ξ
n
: n ∈ Z} stands for the standard Bernoulli sequence and {α
n
: n 1 } is a given
sequence of elements of ℓ
2
Z. Consider a function h ∈ C
2 b
. Then, for Z ∼ N 0, 1,
E[hF] − E[hZ] 6
7 16
× min4
khk
∞
, kh
′′
k
∞
VarG
n
× rX
i ∈Z
α
n i
4
5.55 +
35 24
× kh
′′
k
∞
VarG
n 2
× X
i ∈Z
α
n i
4
with VarG
n
= 3
16 X
i ∈Z
α
n i
2
+ 1
8 X
i ∈Z
α
n i
α
n i+1
. 5.56
It follows that a sufficient condition to have F
n Law
→ Z is that X
i ∈Z
α
n i
4
= o
VarG
n 2
as n
→ ∞. Proof. Identity 5.56 is easily verified. On the other hand, by 5.52, we have
F
n
= G
n
− EG
n
p VarG
n
= J
1
f + J
2
g, with
f =
1 4
p VarG
n
X
a ∈Z
α
n a
1
{a}
+ 1
{a+1}
g =
1 8
p VarG
n
X
a ∈Z
α
n a
1
{a}
⊗ 1
{a+1}
+ 1
{a+1}
⊗ 1
{a}
. Now, let us compute each quantity appearing in the RHS of 5.53. If i
6= j then g ⋆
1 1
gi, j = 1
64VarG
n
α
n i
α
n i+1
1
{ j=i+2}
+ α
n j
α
n j+1
1
{ j=i−2}
. Hence
kg ⋆
1 1
g 1
∆
2
k
ℓ
2
Z
⊗2
= 1
64VarG
n
È X
i, j ∈Z
h α
n i
2
α
n i+1
2
1
{ j=i+2}
+ α
n j
2
α
n j+1
2
1
{ j=i−2}
i
= p
2 64VarG
n
rX
i ∈Z
α
n i
2
α
n i+1
2
. We have
f ⋆
1 1
gi = 1
32VarG
n
α
n i
α
n i+1
+ α
n i
−1 2
+ α
n i
2
+ α
n i
−1
α
n i
−2
.
1729
Hence, using a + b + c + d
2
6 4a
2
+ b
2
+ c
2
+ d
2
, k f ⋆
1 1
g k
ℓ
2
Z
6 1
16VarG
n
rX
i ∈Z
α
n i
2
α
n i+1
2
+ X
i ∈Z
α
n i
−1 4
+ X
i ∈Z
α
n i
4
+ X
i ∈Z
α
n i
−1 2
α
n i
−2 2
= p
2 16VarG
n
rX
i ∈Z
α
n i
2
α
n i+1
2
+ X
i ∈Z
α
n i
4
. We have, using a + b
4
6 8a
4
+ b
4
, X
k ∈Z
f
4
k = 1
256VarG
n 2
X
k ∈Z
X
a ∈Z
α
n a
1
{a}
k + 1
{a+1}
k
4
6 1
16VarG
n 2
X
k ∈Z
α
n k
4
. Finally, still using a + b
4
6 8a
4
+ b
4
, X
k ∈Z
X
i ∈Z
|gi, k|
4
6 1
4096VarG
n 2
X
k ∈Z
X
i,a ∈Z
α
n a
1
{a}
i1
{a+1}
k + 1
{a+1}
i1
{a}
k
4
6 1
256VarG
n 2
X
k ∈Z
α
n k
4
. Now, the desired conclusion follows by plugging all these estimates in 5.53, after observing that
P
i ∈Z
α
n i
2
α
n i+1
2
6 P
i ∈Z
α
n i
4
, by the Cauchy-Schwarz inequality.
6 Multiple integrals over sparse sets
6.1 General results
Fix d 2. Let F
N
, N 1, be a sequence of subsets of N
d
such that the following three properties are satisfied for every N 1: i F
N
6= ;, ii F
N
⊂ ∆
N d
as defined in 2.2, that is, F
N
is contained in {1, . . . , N}
d
and has no diagonal components, and iii F
N
is a symmetric set, in the sense that every i
1
, ..., i
d
∈ F
N
is such that i
σ1
, ..., i
σd
∈ F
N
for every permutation σ of the set {1, ..., d}. Let X
be the infinite Rademacher sequence considered in this paper. Given sets F
N
as at points i–iii, we shall consider the sequence of multilinear forms
e S
N
= [d × |F
N
|]
−
1 2
X
i
1
,...,i
d
∈F
N
X
i
1
· · · X
i
d
= J
d
f
N
, N 1,
6.57 where
|F
N
| stands for the cardinality of F
N
, and f
N
i
1
, ..., i
d
:= [d × |F
N
|]
−
1 2
× 1
F
N
i
1
, ..., i
d
. 1730
Note that Ee S
N
= 0 and Ee S
2 N
= 1 for every N . In the paper [5], Blei and Janson studied the problem of finding conditions on the set F
N
, in order to have that the CLT e
S
N Law
→ Z ∼ N 0, 1, N
→ ∞, 6.58
holds.
Remark 6.1. Strictly speaking, Blei and Janson use the notation F
N
in order to indicate the restric- tion to the simplex
{i
1
, ..., i
d
: i
1
i
2
... i
d
} of a set verifying Properties i–iii above. In order to state Blei and Janson’s result, we need to introduce some more notation.
Remark on notation. In what follows, we will write a
k
to indicate vectors a
k
= a
1
, ..., a
k
belonging to a set of the type
{1, ..., N}
k
=: [N ]
k
, for some k, N 1. We will regard these objects both as vectors and sets, for instance: an expression of the type
a
k
∩ i
l
= ;, means that the two sets {a
1
, ..., a
k
} and {i
1
, ..., i
l
} have no elements in common; when writing j ∈ a
k
, we mean that j = a
r
for some r = 1, ..., k; when writing
a
k
⊂ i
l
k 6 l, we indicate that, for every r = 1, ..., k, one has a
r
= i
s
for some s = 1, ..., l. When a vector a
k
enters in a sum, we will avoid to specify a
k
∈ [N]
k
, whenever the domain of summation [N ]
k
is clear from the context. Given N 1 and an index j
∈ [N], we set F
∗ N , j
= {i
d
∈ F
N
: j ∈ i
d
}. For every N , the set F
N
⊂ F
N
× F
N
is defined as the collection of all pairs i
d
, k
d
∈ F
N
× F
N
such that: a
i
d
∩ k
d
= ;, and b there exists p = 1, ..., d − 1, as well as i
′ p
⊂ i
d
and k
′ p
⊂ k
d
such that
k
′ p
, i
d
\ i
′ p
, i
′ p
, k
d
\ k
′ p
∈ F
N
× F
N
, where
i
d
\ i
′ p
represents the element of [N ]
d −p
obtained by eliminating from i
d
the coordinates belonging to
i
′ p
, and k
′ p
, i
d
\ i
′ p
is the element of [N ]
d
obtained by replacing i
′ p
with k
′ p
in i
d
an analogous description holds for
i
′ p
, k
d
\ k
′ p
. In other words, the 2d indices i
1
, . . . , i
d
, j
1
, . . . , j
d
can be partitioned in at least two ways into elements of F
N
.
Theorem 6.2. [5, Th. 1.7] Let the above notation and assumptions prevail, and suppose that
lim
N →∞
max
j6N
|F
∗ N , j
| |F
N
| = 0,
and 6.59
lim
N →∞
|F
N
| |F
N
|
2
= 0. 6.60
Then Relation 6.58 holds, with convergence of all moments.
Remark 6.3. As pointed out in [5], Condition 6.60 can be described as a weak “sparseness condi- tion” see e.g. [4]. See also [5, Th. 1.7] for a converse statement.
The principal achievement of this section is the following refinement of Theorem 6.2.
1731
Theorem 6.4. Under the above notation and assumptions, consider a function h ∈ C
